Quantitative Tracy–Widom laws for the largest eigenvalue of generalized Wigner matrices

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix H converge to the Tracy–Widom laws at a rate nearly O(N−1∕3), as the matrix dimension N tends to infinity. We allow the variances of the entries of H to have distinct values but of comparable sizes such that ∑iE|hij|2=1. Our result improves the previous rate O(N−2∕9) by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.